Several notions of game enjoy a Nash-like notion of equilibrium without guarantee of existence. There are different ways of weakening a definition of Nash-like equilibrium in order to guarantee the existence of a weakened equilibrium. Nash's approach to the problem for strategic games is probabilistic, \textit{i.e.} continuous, and static. CP and BR approaches for CP and BR games are discrete and dynamic. This paper proposes an approach that lies between those two different approaches: a discrete and static approach. multi strategic games are introduced as a formalism that is able to express both sequential and simultaneous decision-making, which promises a good modelling power. multi strategic games are a generalisation of strategic games and sequential graph games that still enjoys a Cartesian product structure, \textit{i.e.} where agent actually choose their strategies. A pre-fixed point result allows guaranteeing existence of discrete and non deterministic equilibria. On the one hand, these equilibria can be computed with polynomial (low) complexity. On the other hand, they are effective in terms of recommendation, as shown by a numerical example.
Deep Dive into Discrete Nondeterminism and Nash Equilibria for Strategy-Based Games.
Several notions of game enjoy a Nash-like notion of equilibrium without guarantee of existence. There are different ways of weakening a definition of Nash-like equilibrium in order to guarantee the existence of a weakened equilibrium. Nash’s approach to the problem for strategic games is probabilistic, \textit{i.e.} continuous, and static. CP and BR approaches for CP and BR games are discrete and dynamic. This paper proposes an approach that lies between those two different approaches: a discrete and static approach. multi strategic games are introduced as a formalism that is able to express both sequential and simultaneous decision-making, which promises a good modelling power. multi strategic games are a generalisation of strategic games and sequential graph games that still enjoys a Cartesian product structure, \textit{i.e.} where agent actually choose their strategies. A pre-fixed point result allows guaranteeing existence of discrete and non deterministic equilibria. On the one ha
Not all strategic games have a (pure) Nash equilibrium. On the one hand, Nash's probabilistic approach copes with this existence problem with an ad hoc solution: Nash's solution is dedicated to a setting with real-valued payoff functions. On the other hand, CP and BR games propose an abstract and general approach that is applicable to many types of game. Both approaches generalise the notion of Nash equilibrium and guarantee the existence of a weakened Nash equilibrium. There are two main differences between the two approaches though. First, Nash's approach considers finite objects and yields continuous objects, whereas the CP and BR approach preserves finiteness. Second, Nash's approach is static, whereas the CP and BR approach is dynamic: Nash's approach is static because a probabilistic Nash equilibrium can be interpreted as a probabilistic status quo that is (pure) Nash equilibrium of a probabilised game. The Cartesian product structure enables a static approach. CP and BR approach is dynamic because a CP or BR equilibrium can be interpreted as a limit set of states that are tied together by explicit forces. It may be interesting to mix features from both approaches, and to present for instance a discrete and static notion of equilibrium. Actually, such an approach was already adopted in [4], whose purpose was to provide sequential tree games with a notion of discrete non deterministic equilibrium. This approach assumed partially ordered payoffs, and simple "backward induction" guarantees existence of non deterministic subgame perfect equilibrium. This result is superseded by [5] which adopts a completely different approach, but the discrete non determinism spirit can be further exploited.
This paper introduces the concept of abstract strategic games, which corresponds to traditional strategic games where real-valued payoff functions have been replaced with abstract objects called outcomes. In addition, the usual total order over the reals has been replaced with binary relations, one per agent, that account for agent’s preferences over the outcomes. Abstract strategic games thus generalise strategic games like abstract sequential tree games generalise sequential tree games. A notion of Nash equilibrium is defined, but not all abstract strategic games have a Nash equilibrium since traditional strategic games already lack this property.
Like Nash did for traditional strategic games, an attempt is made to introduce probabilities into these new games. However, it is mostly a failure because there does not seem to exist any extension of a poset to its barycentres that is relevant to the purpose. So, instead of saying that “an agent chooses a given strategy with some probability”, this paper proposes to say that “the agent may choose the strategy”, without further specification.
The discrete non determinism proposed above is implemented in the notion of non deterministic best response (ndbr ) multi strategic game. As hinted by the terminology, the best response approach is preferred over the convertibility preference approach for this specific purpose. (Note that discrete non determinism for abstract strategic games can be implemented in a formalism that is more specific and simpler than ndbr multi strategic games, but this general formalism will serve further purposes.) This paper defines the notion of ndbr equilibrium in these games, and a pre-fixed point result helps prove a sufficient condition for every ndbr multi strategic game to have an ndbr equilibrium. An embedding of abstract strategic games into ndbr multi strategic games provides abstract strategic games with a notion of non deterministic (nd ) equilibrium that generalises the notion of Nash equilibrium. Since every abstract strategic game has an nd equilibrium (under some condition), the discrete non deterministic approach succeeds where the probabilistic approach fails, i.e. is irrelevant. This new approach lies between Nash’s approach, which is continuous and static, and the abstract approaches of CP and BR games, which are discrete and dynamic. Indeed, this notion of nd equilibrium is discrete and static. It is deemed static because it makes use of the Cartesian product structure, which allows interpreting an equilibrium as a “static state of the game”.
This paper also defines the notion of multi strategic game that is very similar to the notion of ndbr multi strategic game, while slightly less abstract. multi strategic games are actually a generalisation of both abstract strategic games and sequential graph games. Informally, they are games where a strategic game takes place at each node of a graph. (A different approach to “games network” can be found in [7]) They can thus model within a single game both sequential and simultaneous decision-making mechanisms. An embedding of multi strategic games into ndbr multi strategic games provides multi strategic games with a notion of non deterministic (nd ) equilibrium. In addition, a numerical
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